So I think I need to project one onto the other....
But when I do that I get another vector, that's <144/7,48/7,72/7>. Do I find the magnitude of this, and that should be the distance? I'm lost. The magnitude of that vector is 24, which is a clean answer so it could be right.....

So I think I need to project one onto the other....
But when I do that I get another vector, that's <144/7,48/7,72/7>. Do I find the magnitude of this, and that should be the distance? I'm lost. The magnitude of that vector is 24, which is a clean answer so it could be right.....

You normal vector is wrong. If c is normal (perpendicular) to L1 or L2 then
c.L1 = 0
c.L2 = 0